The title of both this note and the paper it discusses is inspired by a 1970s classic paper by Nobel Laureate Edward Purcell on Life at low Reynolds number. With simple physics, that paper gave insights about micro-organisms (bacteria, sperms etc.) and their locomotion (swimming). We shall discuss it later. This note is about sort of follow-up recent research. We discuss about meso-organisms, cells of plants, with size close to a millimeter - a few orders of scale up from micro-organisms (size in microns) and few orders of size below macro-beasts like us.

Biophysics usually studies biological (bio-chemical) processes by applying physical laws at a micro level - single molecule. Plants such as aquatic Chara (size in centimeters) - known to be around for about 500 million years it seems - have stems in millimeters containing cells (of 0.1 mm dia, few mm length) that are very big for micron sized molecules of biophysics. The plant is shown on the left in the picture below (source).

One of the major reason locomotion of anything is necessary in (and by) organisms is food. Nutrient sensing (called chemoreception, via chemoreceptor proteins) and transport in microorganisms can happen by diffusion. Macro-organisms have a hierarchical transport structure. In us, macro-organisms, things can travel by convection, faster than diffusion - necessary to traverse the body size within short duration. Of course, diffusion is also present but only at the cell level.

In plants of meso-size however, the cells are large enough and diffusion transport transfers nutrients quite slow while pure convection (physical movement of nutrient) is still strongly impeded by viscous damping (Reynolds numbers are quite low, << 1). So, they adopt a combination of diffusion and convection (or advection) strategy to transport nutrients.

One such mechanism is described by the research reported in the 2008 PRL paper (by Jan-Willem van de Meent et al., reference below) I am discussing in this note. The relevant question to ask, as posed in the Physics Viewpoint write-up of the paper by T. Squires, is: How can such a large, single cell drive the steady flows required for life, without the benefit of "macroscopic" moving parts like pumps or muscles?

The short answer is: cyclosis. The establishment of this type of motion and its effects are discussed in the PRL paper.

Cyclosis is a cell-level transport process within such plant cells. A video of cyclosis is here. Let me give a short explanation with the picture below (source).

The Actin filaments (in red) are charged and the myosin-like (motor proteins in large cells) particles (blue dots touching the red filaments) get dragged on the actin as wheels on railroads. They in turn drag the endoplasmic reticulum (ER) nodules attached to them and cause portions of cytoplasm close to the wall to move along (blue arrows in top picture). This movement along the cell wall is along two helical paths - co-axial, one up and one down - as shown in the center part of the first picture (of Chara plant). This movement of fluid inside cell is cyclosis. It is faster than diffusion.

As the authors point out, the advection-diffusion problem of Characean algae is a form of Stokes flow not previously examined. Because, interestingly, the speed U of the helical streaming discussed above can reach in a Chara plant cell of radius R as large as 0.5 mm. Even for smallest molecular species (nutrients etc.), with diffusion constant , the Peclet number . Peclet number, a dimensionless one, determines in a region of length scale R, whether diffusion or advection (convection) dominates. Thus, advection strongly dominates diffusion in the above helical steaming of Chara plant cell, even though the Reynolds number is small (if we assume cell fluid having kinematic viscosity as that of water, ).

In the top Chara plant picture, myosin are shown in green in center inset. They transport nutrients in the form of spherical vesicles (pink) along counter-winding, helical tracks of actin (yellow) that line the cell walls. The transport pulls the viscous liquid (Reynolds number << 1) at the interior of the cell along the walls (the velocity profile is indicated by the black arrows), which helps to mix the fluid perpendicular to the long axis of the plant (right) and move nutrients around the cell interior.

In the paper, the authors present solutions of the Stokes equation on a helical coordinate system they propose for the cyclosis flow, capturing the two counter streaming helical flow. The interesting outcome is, due to the vortical flow along the wall, the axial Taylor dispersion that is present in such diffusion-convection flows is reduced. This dispersion instead of carrying a chunk of chemical intact, stretches - disperses - it due to local convection. The Chara plant cell due to helical motion seem to have found a way to reduce dispersion of nutrients while transporting.

In summary, through their results, the authors show that the combination of helical forcing as presented in [Fig. 1(b)] just above, and high Peclet numbers as we saw earlier, leads to intriguing properties. Termed as "Nature’s microfluidic transporter," by them, enables the Chara plant cell with:

(i) fast radial mass redistribution, (ii) enhanced mass flux across the boundary of the cell, and (iii) homogenization of longitudinal advection. These arise from the generation of transverse flows.

A lucid review of microfluidics by G. M. Whitesides appeared recently in Nature. Micro-channel flows, falling within the purview of microfluidics, are typically viscous boundary layer flows. The cyclosis results discussed can be imitated to increase mass flow rates in such micro-channel flows encountered in microfluidic devices. One more biomimetic possibility for engineering to mimick nature - although I don't know if it is the most efficient way.

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Paper Discussed in this note:

van de Meent JW, Tuval I, & Goldstein RE (2008). Nature's microfluidic transporter: rotational cytoplasmic streaming at high Péclet numbers. Physical review letters, 101 (17) PMID: 18999789 | Download PDF (free)

Other References

• Physics Viewpoint with similar title as this one, discussing the paper - by T. Squires.
• The origins and the future of microfluidics by G. M. Whitesides Nature 442, 368-373 (27 July 2006) | doi:10.1038/nature05058
• Review of Microfluidics by T. M. Squires and S. R. Quake, Rev. Mod. Phys. 77, 977 (2005).